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The program FlϵX (flexural ϵ for xtals) has been developed for a quick, easy and accurate evaluation of the maximum deformation reached in flexible crystals from a simple optical microscope picture. The program takes advantage of computer vision libraries to find the contours of a bent crystal and fit these to semicircles. It can then calculate the theoretical maximum deformation along its long axis using equations from the Euler–Bernoulli beam theory. [ABSTRACT FROM AUTHOR]

This paper presents a new class of metamaterial beams of tunable and multi-band vibration absorption. The metamaterial beam is composed of uniform and periodic beam cells with locally resonant substructure called dual-action vibration absorber, DA. A DA vibration absorber comprising of three locally resonant subsystems, 3-DOF spring-mass-damper subsystems, is utilized to generate frequency stopbands to stop elastic wave propagation. The governing equations of motion for a periodic beam cell are derived. Several distinct mass and stiffness configurations for the metamaterial beam with DA vibration absorber are proposed. The dispersion relations and presence of three frequency stopbands are studied. A finite element method based on Timoshenko beam theory is used to model and analyze the introduced metamaterial beam with DA vibration absorber. The frequency response simulations agree well with the projected stopbands of the developed dispersion relations of the mass and stiffness configurations. The concept of the presented metamaterial beam with tunable and multi-stopbands is promising for wave propagation attenuation and control applications. [ABSTRACT FROM AUTHOR]

Anti-sliding short piles, a novel technique for slope stabilization, have been applied in engineering practices. Nonetheless, a mature structural calculation theory for these piles is still lacking. In this paper, the study presents an internal force solution model for anti-sliding short piles using the finite difference method. By extending the Euler–Bernoulli beam theory and defining boundary conditions, this study develops a set of finite difference equations for computing the structural forces of anti-sliding short piles. Furthermore, this study conducted laboratory model tests on soil landslide cases reinforced with anti-sliding short piles. By comparing the internal forces and deformations of these piles, the test validates the proposed calculation model for anti-sliding short piles. The results suggest that treating the load-bearing and embedded sections as a unified entity during the calculation process, instead of applying continuity conditions separately at the sliding surface as performed in traditional methods, simplifies the complex solving procedure. Moreover, under identical loading conditions, the displacement, bending moment, and shear force data obtained through the finite difference method closely coincide with the measurements from the model tests, confirming the reliability of the anti-sliding short pile calculation model. Additionally, this study demonstrates that reducing the spacing between nodes along the entire anti-sliding short pile, i.e., decreasing the value of the differential segment length 'h', results in more precise computational outcomes. This research offers valuable insights and references for sustainable solutions in the realm of geological disaster prevention and control. [ABSTRACT FROM AUTHOR]

In this paper, analytical and numerical approaches are used to investigate the vibration and non-linear dynamic responses of a bridge pier under two harmonic excitations, caused by aero- and hydro-flows, acting on different portions of the beam. The mechanical system is reduced using the Galerkin method to an ordinary differential equation, and the multiple scale method (MSM) is employed to analyse sub- and super-harmonic resonances. The variation of beam amplitude of vibration caused by a significant effect due to axial force, and two-frequency excitations due to flows are captured. These conduct to the evaluation of nonlinear dynamic behaviour through some frequency-response curves, time history curves of amplitudes vibrations, and phase diagrams presented. In order to gain more features of this system, the discrete general equation is treated numerically. The results show a good accordance between the analytical and numerical solutions. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear dynamic model for rotating beams with elastic boundary is established. The model considers the influence factors such as pre-twisted, setting angle, thermal gradient and geometric nonlinearity. Firstly, according to the Euler–Bernoulli beam theory, the Lagrange function of the rotating beam with elastic constraints is formulated, and a modified Fourier series method is used to solve the linear part to determine the modal function with elastic boundary. Secondly, the modal expansion of the displacements is carried out, and the nonlinear dynamic equations of a rotating pre-twisted beam with elastic boundary are obtained by Lagrange equation. Finally, the multi-scale method is used to solve the nonlinear problem to study the vibration response of the rotating beam with elastic constraints. The accuracy and stability of this method are verified by convergence analysis and comparison with other literatures. After determining the possibility of 2:1 internal resonance of the system, the influence of key system parameters such as rotational speed, spring stiffness and thermal gradient on vibration characteristics under elastic boundary is analyzed. The results show that the system parameters have a significant influence on the nonlinear phenomena of the system and the influence of elastic boundary cannot be ignored. [ABSTRACT FROM AUTHOR]

Purpose: Sandwich structures with well-designed cellular cores exhibit superior shock resistance compared to monolithic structures of equal mass. This study aims to develop a comprehensive analytical model for predicting the dynamic response of cellular-core sandwich structures subjected to shock loading and investigate their application in protective design. Design/methodology/approach: First, an analytical model of a clamped sandwich beam for over-span shock loading was developed. In this model, the incident shock-wave reflection was considered, the clamped face sheets were simplified using two single-degree-of-freedom (SDOF) systems, the core was idealized using the rigid-perfectly-plastic-locking (RPPL) model in the thickness direction and simplified as an SDOF system in the span direction. The model was then evaluated using existing analytical models before being employed to design the sandwich-beam configurations for two typical engineering applications. Findings: The model effectively predicted the dynamic response of sandwich panels, especially when the shock-loading pulse shape was considered. The optimal compressive cellular-core strength increased with increasing peak pressure and shock-loading impulse. Neglecting the core tensile strength could result in an overestimation of the optimal compressive cellular-core strength. Originality/value: A new model was proposed and employed to optimally design clamped cellular-core sandwich-beam configurations subjected to shock loading. [ABSTRACT FROM AUTHOR]

• A nonlinear dynamic model of a large spacecraft with extendible ultra-flexible appendages is proposed. • A piecewise-actual extension strategy is presented. • The effects of the parameters on the natural frequency are investigated. • The limitations of the previous models using Cartesian variables are exhibited. A nonlinear dynamic model of a large spacecraft with extendible ultra-flexible appendages under the perturbation of gravity gradient that can capture large deformation is proposed in this paper. The proposed model is derived accurately based on the Euler-Bernoulli beam theory and Hamilton's principle with non-Cartesian deformation variables (the stretch and transverse deformation). To address the problem of velocity residual, the piecewise-actual (PWA) extension strategy is presented for the first time. Under the proposed extension strategy, the time varying dynamic characteristics of the natural frequency are investigated by numerical methods. Moreover, the influences of the parameters on the stability of the appendages are studied using the eigenvalue method. Finally, to show the accuracy of the proposed nonlinear model, the dynamic response of the proposed model is compared with those of the previous models for different attitude angle moving states. All results are consistent with each other in the case of small deformation, which verifies the correctness of the proposed nonlinear model. The natural frequencies of stretch and transverse deformations decrease gradually with time due to the axial extension of the flexible beam. The stability analysis demonstrates the rapid extension leads to the dynamic instability of the system. Furthermore, the super-harmonic resonance phenomenon is observed when the attitude angle keeps Sun-facing by the proposed model which indicates that the proposed model overcomes the limitations of nonlinear models with Cartesian variables in large deformation. [ABSTRACT FROM AUTHOR]

The focus of this paper is to examine the static and dynamic instabilities of functionally graded beam that contains multiple inclined cracks under the influence of an axial force comprising both static and time-varying harmonic components. The elasticity modulus and mass density of the functionally graded beam are assumed to vary exponentially along its thickness direction. Local stiffness matrix model-based finite element analysis (FEA) is conducted to determine the bending stiffness and tensile stiffness of the section with a crack, and the coupled effect of tensile and bending loadings. Two-node beam elements with three degrees-of-freedom per node are utilized. By combining the Euler–Bernoulli beam theory with Lagrange method, we derive the governing equations that describe the static and dynamic instabilities of a functionally graded beam with multiple inclined cracks. These equations can be solved as eigenvalue problems to obtain the natural frequency and static critical buckling load of the beam. Furthermore, to investigate the dynamic instability of the system, we use the Bolotin method to determine the boundary between the regions of instability and stability based on the same governing equations. By adopting this approach, the study comprehensively investigates the impacts of crack position, inclination angle, and length, as well as elasticity modulus ratio, static and dynamic load factors on both static and dynamic stabilities of a cracked functionally graded beam to gain valuable insights into the stability and performance of cracked functionally graded structures. [ABSTRACT FROM AUTHOR]

Euler beam theory is always used for soil–pile analysis; however, its use is questionable for laterally loaded large-diameter monopiles. In this paper, the solution on the static impedances of monopiles under a combination of axial and horizontal loads was proposed based on three-dimensional continuous medium theory and Timoshenko beam theory. From the analysis results, the use of the Euler–Bernoulli beam theory leads to larger pile-head impedances, and the feature is more significant with lower aspect ratios and stronger pile-bottom constraints. The difference in the impedance from two theories may be up to 10%–43%. Consequently, Timoshenko beam theory is more appropriate for large-diameter monopiles, especially for monopiles with strong bottom constraints (such as rock-socketed monopiles). Meanwhile, the shear force and bending moment at the pile bottom have a considerable effect on the static impedances of large-diameter short monopiles, and only a consideration of the impedance contribution from the soil around the pile is insufficient. Furthermore, both the deflection and the bending moment of the pile increase due to the second-order effect of axial force, but the effect of the axial force on the impedance can be neglected for practical monopile designs. A simple and efficient approach to evaluate the impedance of large-diameter monopiles is desirable for designers, thus a simplified empirical equation for the pile-head static impedance was proposed for ease in calculating each component of static impedances for the large-diameter monopiles of offshore wind turbines (OWTs). [ABSTRACT FROM AUTHOR]

Free flexural vibration of a simply supported sandwich beam on an elastic foundation is the main purpose of the presented investigation. An analytical model of multi-layered beam on elastic foundation has been prepared. The authors submitted an original beam-foundation interaction model which based on variable parameters of the foundation and their influence on the beam response. This explanation leads to the possibility of continuous characterization of the beam-foundation interplay. A nonlinear mathematical function for symmetrical properties of the foundation has been adopted. The frequency equation as a function of geometric and mechanical properties of the beam and the parameters of the elastic foundation was derived using the Galerkin method. The analytical investigation has been divided into two parts: the analysis of elastic foundation with constant and variable properties. The unconventional shape function and the function of deflection have been introduced and employed. Moreover, the finite element analysis has been performed. Sample analytical and numerical calculations have been performed, demonstrating a good concurrence between both models. The difference between analytical and numerical values of the fundamental natural frequency did not exceed 0:5%. [ABSTRACT FROM AUTHOR]

The transportation of seawater on a grand scale via an ultra-large cold-water pipe situated within the context of ocean thermal energy conversion (OTEC) floating installations inherently presents challenges associated with instability and potential malfunction in the face of demanding operational circumstances. This study endeavors to augment the stability and security of cold-water pipe (CWP) operations by scrutinizing their vibrational attributes across diverse boundary configurations. Initially, we invoke Euler–Bernoulli beam theory to forge the analytical framework and proffer a semi-analytical resolution by utilizing the generalized integral transform technique (GITT). Subsequently, we authenticate the convergence and precision of our proposed approach through comparative analysis with extant theories. Our findings underscore the conspicuous influence of boundary conditions on the convergence of transverse displacement. The influence of internal flow on the transverse displacement and the natural frequency manifests substantial variability under different boundary conditions. Significantly, an escalation in the internal flow velocity triggers a concomitant reduction in the natural frequency, ultimately culminating in instability once the critical velocity threshold is reached. Additionally, the reliance of the transverse displacement and the natural frequency on the clump weight at the bottom is markedly pronounced. Our discoveries propose that pipe stability can be ameliorated by adjusting the clump weight at the bottom. Furthermore, the novel insights obtained through our proposed approach can significantly aid in the early-stage design and analysis of CWP. [ABSTRACT FROM AUTHOR]

Purpose: The dynamic characteristics of fluid-conveying pipelines with an initial deflection on nonlinear elastic foundation with local distribution and discrete point arrangement are studied. Methods: Based on the Euler-Bernoulli beam theory, the nonlinear dynamic control equation with initial imperfections is established under the influence of the von Kármán nonlinear effect and initial imperfections. Then by introducing the Dirac delta function and Heaviside function, the mathematical model of locally distributed coupling and discrete point coupling between pipeline and elastic foundation is established. The effects of two different nonlinear elastic foundation layouts, local layout, and discrete point layout, on the dynamic behavior of pipelines are studied by means of a bifurcation diagram, phase diagram, and power spectral density diagram. Results: Through numerical analysis, it is shown that the functionally graded pipeline has very rich dynamic characteristics under different elastic foundation distribution and initial defect effect. Conclusion: The parameters such as the position of the spring, the nonlinear spring stiffness, the length of the support section spring, and the amplitude of the initial defect have a significant effect on the vibration behavior of the pipeline system under different foundation layouts under pulsating flow. [ABSTRACT FROM AUTHOR]

This paper presents a numerical model using the boundary element method for determining the hydrodynamic added mass and added damping of an elastic bridge pier with arbitrary cross-section. Combining the Euler–Bernoulli beam theory with the constant boundary element method, the modal superposition method is used to consider the deformable boundary conditions on the surface of elastic piers to couple the interaction between the elastic pier and water, and the equations for the hydrodynamic added mass and added damping of a general section pier considering the effect of pier-water coupling are derived. The accuracy of the developed model is verified by a benchmark experiment. The developed model is calculated for circular piers and compared with the added mass analytical formulation. The effects of oscillating frequency and structure geometry on the added mass and added damping are further investigated. Results demonstrate that the developed model can be used to solve the hydrodynamic added mass and added damping problems of the elastic bridge pier. Compared to the analytical formula, the developed method incorporates the consideration of added damping in the analysis of the pier-water coupling problem. Oscillating frequency and structure geometry have significant effects on added mass and added damping. [ABSTRACT FROM AUTHOR]

The present study investigates the free vibration behavior of rotating beams made of functionally graded materials (FGMs) with a tapered geometry. The material properties of the beams are characterized by an exponential distribution model. The stiffness and mass matrices of the beams are derived using the principle of virtual energy. These matrices are then evaluated using three different beam theories: Bernoulli–Euler (BE) or Classical Beam Theory (CBT), Timoshenko (T) or First-order Shear Deformation Theory (FSDT), and Reddy (R) or Third-order Shear Deformation Theory (TSDT). Additionally, the study incorporates uncertainties in the model parameters, including rotational velocity, beam material properties, and material distribution. The mean-centered second-order perturbation method is employed to account for the randomness of these properties. To ensure the robustness and accuracy of the probabilistic framework, numerical examples are presented, and the results are compared with those obtained through the Monte Carlo simulation technique. The investigation explores the impact of critical parameters, including material distribution, taper ratios, aspect ratio, hub radius, and rotational speed, on the natural frequencies of the beams is explored within the scope of this investigation. The outcomes are compared not only with previously published research findings but also with the results of 3-Dimensional Finite Element (3D-FE) simulations conducted using ANSYS to validate the model's effectiveness. The comparisons demonstrate a strong agreement across all evaluations. Specifically, it is observed that for thick beams, the results obtained from FSDT and TSDT exhibit a greater agreement with the 3D-FE simulations compared to CBT. It is shown that the coefficient of variation (C.O.V.) of first mode eigenvalue of TSDT, FSDT and CBT are approximately identical for random rotational velocity and discernible deviations are noted in CBT compared to FSDT and TSDT in the case of random material properties. The findings suggest that TSDT outperforms FSDT by eliminating the need for a shear correction coefficient, thereby establishing its superiority in accurately predicting the natural frequencies of rotating, tapered beams composed of FGMs. [ABSTRACT FROM AUTHOR]

This article investigates the dynamic nonlinear response of three-dimensional structures using variable-kinematics finite beam elements obtained with the Carrera Unified Formulation. The formalism enables one to consider the three-dimensional form of displacement–strain relations and constitutive law. The deformation mechanisms and the associated couplings are described consistently with the selected kinematic model. The Hilbert–Hughes–Taylor method and the iterative Newton–Raphson scheme are adopted to solve the motion equations derived in a total Lagrangian scenario. Various models have been obtained by using Taylor- and Lagrange-like expansions. The capabilities of the beam elements are assessed considering isotropic, homogeneous structures with compact and thin-walled sections. [ABSTRACT FROM AUTHOR]

Herein, an inertial noncontact piezoelectric rotary energy harvester with linear reciprocating motion (L‐PREH) is presented. The existing piezoelectric rotary harvester employing gravity excitation has limited performance when the rotation speed is high due to the negative influence of centrifugal force. L‐PREH translates rotational motion into linear motion via the transmission chain and employs inertial force excitation to overcome high‐speed performance limitations. Using the Euler–Bernoulli beam theory, the motion governing equations of piezoelectric transducers have been derived, and an electromechanical coupling model has been constructed. Moreover, the piezoelectric transducer is simulated and analyzed. The control variable approach is used to explore the key parameters impacting output performance. When the mass is positioned in symmetrical method, the guide rod is fixed in noncentral place, the limiter is fixed in longest distance, the distance between the mass's center and the main frame is the maximum, and the rotating speed is 450 RPM, the maximum peak‐to‐peak output voltage of an L‐PREH single transducer is 24 V. The highest power of two piezoelectric transducers linked in parallel with a load resistance of 400 kΩ is 0.27 mW, which can light up more than 70 light‐emitting diodes. The L‐PREH can drive low‐power devices. [ABSTRACT FROM AUTHOR]

We present a brief overview of the theory of high-intensity nonlinear diffracting beams. Characteristic distortions of the profiles of acoustic waves, which are observed during the wave propagation, are described. The following features are pointed out. First, the positive and negative half periods of the original harmonic signal are differently distorted. The positive-pressure phase duration is shortened and its "amplitude" is increased. On the contrary, the region of negative pressure is somewhat extended and reduced in "amplitude." Second, the profiles are shifted to the region of negative values of the "accompanying" time, i.e., the diffraction of a convex beam leads to a slight increase in its propagation velocity. In addition, the positive pressure in some range of distances may exceed the initial value. Low-frequency geometric dispersion leads to differentiation of the weak signal profile in the focal region and in the far zone, which leads to the disappearance of unipolar video pulses. A stationary wave composed of sections of a parabolic shape can be formed in the waist. The limiting values of acoustic pressure and wave intensity in the focus are estimated. Approximate mathematical methods and the models used in the calculation of the wave profiles are described. [ABSTRACT FROM AUTHOR]

In the context of nonclassical continuum mechanics, the nonlocal strain gradient theory is employed to develop a nonclassical size dependent model to investigate the dynamic behavior of a CNTs reinforced composite beam resting on two parameters elastic foundations under a moving load. The governing dynamic equations of motion are derived based on third-order shear deformation theory using Hamilton's principle. An analytical solution methodology is developed using Navier's procedure is developed to derive the analytical solution for the equations of motion. The developed methodology is checked and compared. Parametric studies are conducted to demonstrate the applicability of the developed procedure to investigate the dynamic behavior of CNTs beams under moving load. Effects of the elastic foundation parameters, volume fraction, CNTs configuration, the nonclassical parameters, and the moving load velocity parameter on the dynamic behavior of CNTs beams under moving load are investigated and analyzed. The obtained results are supportive for design and manufacturing of composite CNTs beams. [ABSTRACT FROM AUTHOR]

To significantly reduce the amplitude of longitudinal vibrations of the beam in the vicinity of its natural frequencies, a fundamentally new method of damping vibrations is used. For this purpose, the beam surfaces are covered with layers of polarized piezoceramics with a strong piezoelectric effect. We will use two types of electrical conditions on the electrodes of the piezoelectric layers: short-circuited electrodes and disconnected electrodes. On short-circuited electrodes, the electric potential is zero. As a result of the piezoelectric effect, an electric charge appears on the disconnected electrodes when the beam is deformed. The electroelastic state of a beam with different electrical conditions is described by different boundary value problems. A new approach to damping vibrations in the vicinity of natural frequencies is based on the following rule for controlling the dynamic characteristics of a structure: when the beam vibration frequency approaches its natural vibration frequency, we change the electrical conditions on the electrodes of the piezoelectric layers, thereby changing the spectrum of its natural frequencies. Let, for example, the vibration frequency of a beam with short-circuited electrodes approach its natural frequency. In this case, the amplitudes of the sought quantities grow without limit. The natural frequency spectrum of a beam with disconnected electrodes will differ from the spectrum of a beam with short-circuited electrodes. As a result, the amplitudes of the sought quantities will decrease. It is shown that the efficiency of vibration damping can be significantly increased by choosing the direction of the preliminary polarization of the piezoelectric material and the location of its electrodes. Numerical examples are given that demonstrate the effectiveness of the proposed method. The advantage of the method lies in its simplicity and the low cost of the piezoelectric material, which serves as a non-inertial damper. [ABSTRACT FROM AUTHOR]

Shape memory polymers (SMP) and its composites (SMPC) have gained popularity for their potential in intelligent structures fields. Developing a reliable mechanical model to accurately describe mechanical behavior of its intelligent structures is important for both theory and practice. This paper presents a novel mechanical model for describing the mechanical behavior of short glass fiber/polyurethane (GF/PU) composite beams, based on a constitutive model of SMP and incorporating principles from composite mechanics theory, the Hamiltonian principle and Euler–Bernoulli beam theory. The model includes kinematic equations and material parameter equations. Based on the constitutive equations of SMP, coupled with Euler–Bernoulli beam theory and Hamilton's principle, the kinematic equations of GF/PU composite beam are established. Additionally, a correction factor was proposed and employed to modify the material parameter equation of SMP, which was then combined with the mechanics theory of composite materials to establish a material parameter equation for composite beams. Taking a simply supported beam as an example, we solve the model using Fourier method and simulate and analyze the material parameters properties, static and viscous mechanical behaviors properties, and shape memory behavior properties of the GF/PU composite beam. The results show that the GF volume fraction, temperature correction coefficient, regularization parameter, and contact parameter have a significant impact on the material parameters of GF/PU composite materials and the material parameters of composite beams have important effects on their mechanical properties. By adjusting material parameters, ideal mechanical properties can be obtained for the GF/PU composite beam. This study provides a theoretical foundation for intelligent beam design and analysis based on SMP and SMPC. Highlights: A novel viscoelastic mechanical model for GF/PU composite beam is proposed.Effect of composite components on GF/PU composite is studied.Kinematic equations based on Hamiltonian principle is established.Material parameter equation including corrected parameter is established. [ABSTRACT FROM AUTHOR]

A new size-dependent axially functionally graded (AFG) micro-beam model is established with the application of a reformulated strain gradient elasticity theory (RSGET). The new micro-beam model incorporates the strain gradient, velocity gradient, and couple stress effects, and accounts for the material variation along the axial direction of the two-component functionally graded beam. The governing equations and complete boundary conditions of the AFG beam are derived based on Hamilton's principle. The correctness of the current model is verified by comparing the static behavior results of the current model and the finite element model (FEM) at the micro-scale. The influence of material inhomogeneity and size effect on the static and dynamic responses of the AFG beam is studied. The numerical results show that the static and vibration responses predicted by the newly developed model are different from those based on the classical model at the micro-scale. The new model can be applied not only in the optimization of micro acoustic wave devices but also in the design of AFG micro-sensors and micro-actuators. [ABSTRACT FROM AUTHOR]

Specifically, the soil displacement in the I z i -direction ( I u SB r sb i ) was described as follows: (1) HT ht where I SB z sb i = dimensionless decay functions of the displacement components in the I z i -direction. The radial and circumferential soil displacements in the lateral direction obtained from T-B with I u SB z sb i and rigid beam with I u SB z sb i are similar, but the deflection diminishes much more slowly if I u SB z sb i is not considered. Comparing the results obtained from T-B with I u SB z sb i with the FDM results, the writers found that the radial, circumferential, and vertical soil displacements obtained from T-B with I u SB z sb i are in better agreement with the FDM results. [Extracted from the article]

(4) and (5) along with the differential equations of I SB z sb i and I SB x sb i as follows: (7) HT ht (8) HT ht where the parameters I i SB 1 sb - I i SB 4 sb describe the rates of the variations of I u SB sz sb i and I u SB sx sb i with depth. [Extracted from the article]

In order to improve the vibration reduction performance of damping treatments, a new damping structure consisting of a uniform base layer and two periodically alternating free layers was examined in this study. Closed-form solutions for both the band structure and the forced response of the periodic bi-layer beam were theoretically derived and verified via numerical solutions using the finite-element method. The results showed that the structure with periodic free-layer damping (PFLD) treatment reduced broadband vibrations, and the levels of reduction were dominated by Bragg scattering in the band gaps and damping in the passbands. The vibration experiment verified the derived theory's accuracy and showed that the PFLD treatment could increase vibration reduction levels in low-frequency band gaps compared with traditional free-layer damping treatments. The effects of the parameters—cell lengths, sub-cell-length ratios, and thickness ratios—were also discussed, providing further understanding of the vibration reduction performance of the bi-layer beam with the PFLD treatment, and this can be used to help designers optimize the periodic bi-layer beam to achieve better performance. [ABSTRACT FROM AUTHOR]

Structures with inhomogeneous materials, non-uniform cross-sections, non-uniform supports, and subject to non-uniform loads are increasingly common in aerospace applications. This paper presents a simple and unified numerical dynamics model for all beams with arbitrarily axially varying cross-sections, materials, foundations, loads, and general boundary conditions. These spatially varying properties are all approximated by high-order Chebyshev expansions, and discretized by Gauss–Lobatto sampling. The discrete governing equation of non-uniform axially functionally graded beams resting on variable Winkler–Pasternak foundations subjected to non-uniformly distributed loads is derived based on the Euler–Bernoulli beam theory. A projection matrix method is employed to simultaneously assemble spectral elements and impose general boundary conditions. Numerical experiments are performed to validate the proposed method, considering different inhomogeneous materials, boundary conditions, foundations, cross-sections, and loads. The results are compared with those reported in the literature and obtained by the finite element method, and excellent agreement is observed. The convergence, accuracy, and efficiency of the proposed method are demonstrated. [ABSTRACT FROM AUTHOR]

Purpose: In this paper, the transverse vibrations of an axially moving cantilever beam subject to a continuously distributed moving mass are studied numerically. Methods: An elastic coupling coefficient is introduced to describe the actual elastic coupling effect between the beam and moving mass. The motion equations of the system are derived by Lagrange's equation and Galerkin method. The Newmark-beta direct time integrating method is adopted to analyze the dynamic responses. Results and Conclusion: The motion equations are verified by comparing the dynamic responses with previous literature. An interesting energy separation phenomenon is observed when the moving mass separates from the beam. The effects of moving mass parameters (moving mass velocity, length, and elastic coupling coefficient) on beam dynamics and the energy separation phenomenon are discussed. It has been observed that the elastic coupling effect between the beam and moving mass has a significant effect on beam dynamics. [ABSTRACT FROM AUTHOR]

Purpose: The main aim of this work is to investigate the free vibration of functionally graded rotating nanotubes using the Euler–Bernoulli beam theory under the assumptions of Eringen's nonlocal elasticity theory, which gets close to the nanostructure's behaviors. The material properties are assumed to be graded in the nanotube thickness direction according to the power-law distribution. Methods: Hamilton's principle is used to derive equations of motion and boundary conditions, which are uncoupled and solved analytically using the state-space method. The gyroscopic effect is considered in the equations of motion and, unprecedentedly, in the boundary conditions. Results: The effects of rotating speed, mode number, material power-law index, and geometrical and nonlocal parameters on both critical speeds and natural frequencies under various boundary conditions for functionally graded rotating nanotubes are plotted and investigated for the first time. Conclusion: The current formulation of the rotating FG-NT is expected to serve as a standard for assessing the accuracy of designing various nanomotors, whether experimental, numerical, or analytical. [ABSTRACT FROM AUTHOR]

Purpose: In this paper, free vibration of polynomial and exponential axially functionally graded (AFG) beam with lumped end masses and elastic supports is studied within the Euler–Bernoulli beam theory. The axial eccentricity and rotatory inertia of the end lumped masses are considered. Also, a new equation is proposed for the equivalence of the exponential AFG beams with the polynomial AFG beams. Methods: Both the geometrical and material properties of the beam are graded along the AFG beam axis according to the polynomial (P-AFG) and exponential (E-AFG) functions. An analytical approach to derive the exact characteristic equations of two types of AFG beams with end lumped masses and elastic supports is presented. Accordingly, through numerical solving of the characteristic equations, the exact natural frequencies of AFG beams with arbitrary boundary conditions are obtained. Results and Conclusion: The effects of end lumped mass parameters, i.e., end mass ratio, rotatory inertia ratio, axial eccentricity ratio, the AFG parameters, and the end support parameters on the first three natural frequencies of several P-AFG beams and their equivalent E-AFG beams are investigated. Results show the mentioned parameters play an important role in determining the natural frequencies for the AFG beams. Moreover, the present results can use as a benchmark for other numerical solutions and be served for purposeful design vibrating a wide range of non-uniform and composite beams. [ABSTRACT FROM AUTHOR]

An analytical wave propagation approach is developed in this paper for the free and forced vibration of a system of multiple elastically connected beams for the first time. The beams of the system are continuously joined by a massless, linear, elastic layer which can be regarded as continuous spring. The coupled partial differential equations governing the vibration of the multi-beam system are established and decoupled by using a technic developed based on matrix theory. For the decoupled equations, a general "vibration" state is introduced into the symplectic dual system. By solving the symplectic eigenproblem and utilizing the wave propagation theory, the general "vibration" state can be analytically described in symplectic space. By using these analytical expressions and satisfying the physical boundary conditions of the system, the natural frequencies, mode shapes and forced responses can be obtained analytically and explicitly. In the numerical examples, free and forced transverse vibration of the two- and three-beam system with various combinations of boundary conditions are considered. The effectiveness of the present method is validated by comparing the present results with the analytical results from the literature and the results calculated by the finite element method. [ABSTRACT FROM AUTHOR]

Variable-cross-section beams have better mass and strength distribution compared with constant cross-section beams, which can optimize the harvesting power of piezoelectric vibration energy harvesters, which are widely used in self-supplied and low-power electronic devices, providing more convenience and innovation for the development of micromechanical intelligence and portable mobile devices. This paper proposes a piezoelectric energy harvester with a tristable-exponential-decay cross section, which optimizes the strain distribution of the cantilever beam through exponential decay changes to improve the harvesting efficiency of the harvester in low-frequency environments. First, the nonlinear magnetic force is obtained based on the magnetic dipole, and the dynamic model is established by using the Euler–Bernoulli beam theory and Lagrangian equation. The influence of the structural parameters of the harvester on the system dynamics and output characteristics is analyzed in the two dimensions of time and frequency, and the influence of the exponential decay coefficient on the system dynamic response and output power is deeply studied. The research shows that the exponential decay section can reduce the first natural frequency of the cantilever beam; by changing the amplitude, frequency, d and dg of the excitation acceleration, the switching of the monostable, tristable and bistable states of the system can be realized. With a decrease in the exponential decay coefficient, under a low-frequency excitation of 0–7 Hz, the output power of the cantilever beam per unit volume is significantly improved, and under a 4 Hz low-frequency excitation, the acquisition output power per unit volume of the cantilever beam is increased by 7 times, thus realizing low-frequency, high-efficiency energy harvesting. [ABSTRACT FROM AUTHOR]

Extensive prior research has delved into the localization of elastic wave energy through defect modes within phononic crystals (PnCs). The amalgamation of defective PnCs with piezoelectric materials has opened new avenues for conceptual innovations catering to energy harvesters, wave filters, and ultrasonic receivers. A recent departure from this conventional paradigm involves designing an ultrasonic actuator that excites elastic waves. However, previous efforts have mostly focused on single-defect scenarios for bending-wave excitation. To push the boundaries, this research takes a step forward by extending PnC design to include double piezoelectric defects. This advancement allows ultrasonic actuators to effectively operate across multiple frequencies. An analytical model originally developed for a single-defect situation via Euler–Bernoulli beam theory is adapted to fit within the framework of a double-defect set-up, predicting wave-excitation performance. Furthermore, a comprehensive study is executed to analyze how changes in input voltage configurations impact the output responses. The ultimate goal is to create ultrasonic transducers that could have practical applications in nondestructive testing for monitoring structural health and in ultrasonic imaging for medical purposes. [ABSTRACT FROM AUTHOR]

Peridynamics is a continuum theory that operates with non-local deformation measures as well as long-range internal force/moment interactions. The resulting equations are of the integral type, in contrast to the classical theory, which deals with differential equations. The aim of this paper is to analyze peridynamic governing equations for elastic beams. To this end, the strain energy density is formulated as a function of the non-local curvature. By applying the Lagrange principle, the peridynamic equations of motion are derived. Examples of non-local boundary conditions, including simple support, clamped edge, roller clamped edge, and free edge, are presented by introducing the interaction domain. Novel closed-form analytical solutions to integral equations are presented for beams with various boundary conditions, including clamped—simply supported, clamped–clamped, simply supported–roller-clamped, and clamped–roller-clamped beams. Furthermore, different types of loadings, including uniformly distributed load, concentrated force, and concentrated moment, are considered. The results are validated by comparing the derived solutions against solutions to the classical Bernoulli–Euler beam theory. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes, which shows the capability of the derived equations of motion and proposed boundary conditions. [ABSTRACT FROM AUTHOR]

The nonlinear, in-plane mechanics of a thin-walled honeycomb with zero Poisson's ratio under large deformation is investigated in this paper. A theoretical method for calculating in-plane tensile modulus, modified factors of linear constitutive relations of the honeycomb structures with zero Poisson's ratio is proposed based on the theory of Euler-Bernoulli beam and the bending theory of beam in large deflection, and a finite element simulation is given to validate. In addition, parametric analysis for revealing the impacts of geometrical configurations and material parameters on in-plane mechanical properties of the honeycombs have been studied systematically. These findings suggest that geometric and/or material parameters provide different contributions to the effective mechanical properties and lead to a separate design for the in-plane mechanical properties. After that, the effects of geometric and/or material nonlinearities on mechanical properties of the honeycomb structures with zero Poisson's ratio are revealed by considering the dimensionless tangent stiffness of the honeycombs. [ABSTRACT FROM AUTHOR]

In this paper, non-linear free vibration analysis of nano-beam has been studied. The non-local strain gradient theory and curvature tensor are used to show the size effect. The length scale parameter expresses the effect of strain gradient tensor in the non-local strain gradient theory. However, the aim of this article is to show the simultaneous effect of curvature and strain gradient tensors in non-linear vibration of functionally graded porous nano-beams. The effect of curvature tensor is demonstrated with the curvature tensor dependent parameter. Considering non-linear Von Kármán strains and Euler–Bernoulli beam theory, the governing vibrational equation of FG porous nano-beams are derived using Hamilton's principle in the presence of strain gradient and curvature tensors simultaneously. The non-linear differential equation is extracted by using Galerkin's method and the non-linear natural frequency of nano-beam is obtained according to Hamiltonian approach. Results represent the simultaneous effects of the length scale and curvature tensor dependent parameters on dimensionless non-linear natural frequencies. Also effects of different parameters such as non-local parameter, length scale parameter, porosity volume index, and power-law index are discussed in the presence and absence of the curvature tensor dependent parameter. Also, the beginning points of stiffness-hardening and stiffness-softening of nano-beam are always constant values in the non-local strain gradient theory, whereas considering the curvature tensor changes the beginning points of stiffness-hardening and stiffness-softening. The results are also compared with previous researches for validation. [ABSTRACT FROM AUTHOR]

This paper compares the accuracy of the integral equation method and the multiple scales method in solving the amplitude equation of multi degree of freedom nonlinear vibration systems. We consider three examples: a two-degree-of-freedom stainless-steel beam system controlled by a saturation controller, a three-degree-of-freedom rotating compressor blade model and a four-degree-of-freedom horizontally supported Jeffcott-rotor system controlled by a PPF controller. The amplitude equations are obtained by applying the integral equation method and the method of multiple scales. The stability analysis is achieved based on the Floquet theory together with Routh–Hurwitz criterion. Furthermore, we modified the iterative procedure of the integral equation method to make the analytic approximate solution more accurate. Finally, the analyses show that in most cases, the analytical solutions obtained by the integral equation method are more excellent agreement with the numerical solutions than the most commonly used method of multiple scales. Therefore, the integral equation method is worth popularizing to obtain the approximate analytical solutions of the multi-degree-of-freedom nonlinear vibration system. [ABSTRACT FROM AUTHOR]

Using a three-variable higher-order shear deformation theory (HSDT), this research proposes an analytical method for studying the free vibration and stability of perfect and imperfect functionally graded (FG) beams resting on variable elastic foundations (VEFs). Unlike the other HSDTs, in this study, the number of unknown functions involved is only three, while the other HSDTs include four unknown functions. Besides, this theory meets the boundary requirements of zero tension on the beam surfaces and allows for hyperbolic distributions of transverse shear stresses without the need for shear correction factors. The elastic medium is supposed to have two parameters (i.e., Winkler–Pasternak foundations), with the Winkler parameter in the longitudinal direction being variable variations (linear, parabolic, sinusoidal, cosine, exponential, and uniform) and the Pasternak parameter being fixed, at first.1 The effective material characteristics of the FG beam are assumed to follow a simple power-law distribution in the thickness direction. Furthermore, the influence of porosity is investigated by considering four distinct types of porosity distribution patterns. First, the equations of motion are derived using Hamilton's principle, and then Navier's method is used to solve the system of equations for the FG beam with simply supported ends analytically. The correctness of the current formulation is demonstrated by comparing them with the results of open literature. Finally, parametric studies are done to explore the impacts of various parameters on the free vibration and buckling behaviors of FG beams. The new theory is shown to be not only correct but also simple in predicting the free vibration and buckling responses of FG beams resting on VEFs. [ABSTRACT FROM AUTHOR]

It has been well established that the internal length scale related to the cell size plays a critical role in the response of architected structures. It this paper, a Volterra derivative-based approach for deriving nonlocal continuum laws directly from an energy expression without involving spatial derivatives of the displacement is proposed. A major aspect of the work is the introduction of a nonlocal derivative-free directionality term, which recovers the classical deformation gradient in the infinitesimal limit. The proposed directionality term avoids issues with correspondences under nonsymmetric conditions (such a unequal distribution of points that cause trouble with conventional correspondence-based approaches in peridynamics). Using this approach, we derive a nonlocal version of a shear deformable beam model in the form of integro-differential equations. As an application, buckling analysis of architected beams with different core shapes is performed. In this context, we also provide a physical basis for the consideration of energy for nonaffine (local bending) deformation. This removes the need for additional energy in an ad hoc manner towards suppressing zero-energy modes. The numerical results demonstrate that the proposed framework can accurately estimate the critical buckling load for a beam in comparison to 3-D simulations at a small fraction of the cost and computational time. Efficacy of the framework is demonstrated by analysing the responses of a deformable beam under different loads and boundary conditions. [ABSTRACT FROM AUTHOR]

Given their considerable specific surface area and amorphous characteristics, nanoparticles exhibit excellent pozzolanic activity, and when undergoing a reaction with calcium hydroxide, this leads to the generation of a denser matrix by promoting the formation of a greater amount of C-S-H gel, thereby enhancing the strength and durability of the concrete and fortifying the overall structure. Indeed, the present study investigates a comparative study of the buckling and free vibration analyses of concrete beams reinforced with various types of nanoparticles. For its simplicity and accuracy, a higher-order shear deformation theory will be used to analytically model the reinforced concrete beam. Furthermore, the powerful Eshelby's model is used to derive the equivalent nanocomposite properties. The soil medium is simulated with Pasternak elastic foundation, including a shear layer, and Winkler's spring, interlinked with a Kerr foundation. The motion equations are derived using Hamilton's principle. Moreover, based on Navier's analytical methods, the closed-form solutions of simply supported beams have been obtained. Different parameters, such as type and volume percent of nanoparticles, geometrical parameters, choice of theory and soil medium, on the buckling and dynamic behavior of the beam, are exercised and shown. The major findings of this work indicate that the use of nanoparticles in concretes increases better mechanical resistance and amplifies the natural frequencies. In addition, the elastic foundation has a significant impact on the buckling and vibration performances of concrete beams. [ABSTRACT FROM AUTHOR]

Beam selection aims to select beams with large power and little inter‐user interference (IUI). In wireless communication scenarios, the levels of IUI are usually different, but, most existing beam selection methods apply the same beam selection criteria for all users, which is detrimental to maintain an attractive trade‐off between computational complexity and system performance. Considering the discrepancy of IUI comprehensively, this paper proposes a beam selection method based on interfering user grouping (IUG) for wideband millimetre‐wave massive multiple‐input multiple‐output (MIMO) systems. Firstly, a candidate beam set (CBS) is constructed in light of the sparsity and power distribution of the beamspace channel. Based on the CBS, all users are classified into non‐interfering users (NIUs), low‐interference users (LIUs) and high‐interference users (HIUs). For NIUs, the beams with large power are selected from CBS under the magnitude maximization (MM) criterion; for LIUs, a novel MM criterion with a tabu list is designed, which can effectively tackle the drawback of assigning shared beams in traditional MM; for HIUs, the incremental algorithm based on the sum‐rate maximization criterion is developed to find the optimal beams from CBS. Simulation results validate that the proposed IUG‐based method can obtain a good performance with lower computational complexity compared with the existing methods. [ABSTRACT FROM AUTHOR]

For the first time, a two-node beam element with nine degrees of freedom (DOFs) was developed in this paper using a combination of modified couple stress and k th-order shear deformation beam theory to analyze the free vibration and static buckling responses of rotating bidirectional functionally graded porous (BDFGP) microbeams resting on variable elastic foundations, in which the whole structure was exposed to a magnetic field in the hygrothermal condition. The mechanical properties and the length-scale parameters varied in both the thickness and length directions, which in the novel porosity model and power-law indexes of the material were considered to be a function of the x - and z -coordinates. The present method's accuracy was determined by comparing it with published findings. Additionally, extensive research was conducted to evaluate the effect of various parameters on the mechanical responses of the rotating BDFGP microbeam. The numerical results showed that porosity coefficient, elastic foundation parameters, and hygrothermal environment all significantly affect the rotating BDFGP microbeam's free vibration and static buckling behavior. [ABSTRACT FROM AUTHOR]

The classical Euler–Bernoulli beam model and Kirchhoff plate model are very useful on the macroscopic scale. In the context of Eringen's nonlocal elasticity theory, this article aims to develop the criteria of the applicability of nanoscale Euler–Bernoulli beam and Kirchhoff plate models based on the Timoshenko beam model and the Mindlin plate model via the wave propagation theory. The corresponding governing differential equations for the nanoscale Timoshenko beam and Mindlin plate are derived by the Hamilton's principle, and the dispersion equations of wave are then obtained. By applying Taylor expansion to the corresponding solutions of the dispersion equations, new criteria are developed, simultaneously taking into account the effects of nonlocal parameters and material properties. When the nonlocal parameter is set to zero, the present criteria may be readily degenerated to their macroscopic counterparts. According to the present criteria, this article systematically evaluates the existing studies in literature. Various works in literature did not consider the effect of the nonlocal parameter, and hence, failed to satisfy the application conditions of the Euler–Bernoulli beam and Kirchhoff plate models on the nanoscale. The work in this article is of scientific significance to various studies on nanostructures. [ABSTRACT FROM AUTHOR]

The seismic capacity of wind turbine support towers is of significant concern as wind power provides an increasing proportion of the world's electricity supply. This study presents a computational study on the seismic performance of steel-concrete hybrid towers (SCHTs). The equations that govern the tower-free vibration responses are derived based on Euler-Bernoulli beam theory. The modal results are used in the response spectrum analysis to evaluate the higher-mode effects in the SCHTs. Then, a cantilever beam model capable of capturing the joint opening and closing was developed for structural analyses and calibrated against finite element models. Finally, dynamic time history analyses were conducted for different SCHTs under far-field (FF) and near-fault (NF) earthquakes. These analyses showed that the second mode of SCHTs is more significant for the shear force diagram. Dynamic amplification causes the mean peak base moment from the FF set and NF set to be 1.30–1.45 and 1.37–1.57, respectively, greater than the design spectrum using the same 5% damping. [ABSTRACT FROM AUTHOR]

This paper concerns with the stabilization of an axially moving beam by an adaptive boundary control. We prove existence and uniqueness of the solution by means of nonlinear semigroup theory. Moreover, we construct the control through a low-gain adaptive velocity feedback. We also prove that the designed control is able to stabilize exponentially the closed loop system. Some numerical simulations are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this study, free vibration of stepped beam which is parallel to a uniform beam with same length and elastically connected to it, is considered. Euler-Bernoulli beam theory has been applied to drive equations of motion, abrupt change in height of beam considered as step and Winkler-type elastic layer model serve as connection between beams. The differential transform method (DTM) is applied to determine dimensionless frequencies and mode shapes. In the case of two uniform parallel beams accuracy of solution is verified by comparing with results reported by other methods. It is assumed all supports have one type and fully clamped and fully hinged supports considered for boundary conditions. The effects of different parameters such as: step location and ratio, connecting layer coefficient and boundary conditions on dimensionless frequencies and mode shapes investigated and discussed. This problem handled for first time in present study and results are completely new. [ABSTRACT FROM AUTHOR]

Purpose: The feasibility of harvesting electrical energy from mechanical vibration is demonstrated in the thesis. In the technique, energy is harvested from simply supported beam vibration under a moving mass using a thin piezoelectric material. Design/methodology/approach: The structure is represented by a basic beam of length L that is supported at both ends and traversed by a moving mass M travelling at a constant velocity v. The Euler-Bernoulli differential equation describes its behaviour. The dynamic analysis of a beam is performed by using three moving masses of (35.61, 65.81, and 79.41) gr each travelling three uniform speeds of (1.6, 2 and 2.4) m/s. A differential equation of the electromechanical system is obtained by transforming the piezoelectric constitutive equation and solved numerically by MATLAB. Findings: The results indicate that the numerical and experimental values for the midpoint deflection of the beam and the piezoelectric voltage are very close. Research limitations/implications: Using the COMSOL programme, the proposed approach is checked by comparing results with data obtained by the finite element method (FEM). An experimental setup was also built and constructed to determine the voltage created by the piezoelectric patch and the beam response as a result of the mass travelling along the beam. Practical implications: The results show that the dynamic deflection, piezoelectric voltage, and piezoelectric energy harvesting all increase as the speed and magnitude of the moving mass increase. The harvesting power vs. load resistance curve begins at zero, increases to a maximum value, and then remains almost constant as the resistance is increased further. The optimal length of the piezoelectric patch was obtained to be 0.63 m. When the length of the beam increases, the resonant frequency decreases, and at the same time the harvested energy increases. However, increasing the beam thickness has the opposite effect; whereas raising the beam width does not affect the resonant frequency but decreases energy harvesting. Originality/value: The most essential point here is the need to have correctly built scale models. They can provide a substantial amount of information at a low cost, accommodate a variety of test settings, and aid in the selection and verification of the most effective analytical model to resolve the actual issue. [ABSTRACT FROM AUTHOR]